Mathematics

Evaluate the given integral.
$$\displaystyle \int { \cfrac { { x }^{ 2 } }{ 1+{ x }^{ 3 } }  } dx$$


SOLUTION
Let $$t=1+{x}^{3}\Rightarrow\,dt=3{x}^{2}dx$$

$$\Rightarrow\,\dfrac{dt}{3}={x}^{2}dx$$

$$I=\displaystyle\int{\dfrac{{x}^{2}dx}{1+{x}^{3}}}$$

$$=\dfrac{1}{3}\displaystyle\int{\dfrac{dt}{t}}$$

$$=\dfrac{1}{3}\log{\left|t\right|}+c$$    ........where $$c$$ is the constant of integration.

$$=\dfrac{1}{3}\log{\left|1+{x}^{3}\right|}+c$$    ............where $$t=1+{x}^{3}$$
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Subjective Medium Published on 17th 09, 2020
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